Space time mesh refinement methods

In practical applications, one may have to treat complex geometries or geometrical details in diffraction problems. In such situations, it is desirable to use local mesh refinements. Moreover, it is highly desirable to be able to treat non matching grids (this is even needed if one wants to use FDTD like scheme in each grid). A first idea consists in using only spatial refinement (see [15] for acoustic waves, [28] and [115] for Maxwell’s equations). However, when a uniform time step is used, it is the finest mesh that will impose the time step because of the stability condition. There are two problems with this: (1) the computational costs will be increased and (2) the ratio cΔt/h on the coarse grid will be much smaller than its optimal value, which will generate dispersion errors. A way to avoid these problems is to use a local time step Δt, related to h in order to keep the ratio cΔt/h constant. This solution however raised other practical and theoretical problems that are much more intricate than in the case of a simple spatial refinement, in particular in terms of stability.